Perturbation theory first-order corrections

The perturbation-theory first-order energy correction is (see also Problem 8.35)... 

A Perturbation Theory is developed for treating a system of n electrons in which the Hartree-Fock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy and the charge density of the system is zero. The expression for the second order correction for the energy greatly simplifies because of the special property of the zero order solution. It is pointed out that the development of the higher order approximation involves only calculations based on a definite one-body problem. 

Up to this point we are still dealing with undetermined quantities, energy and wave funetion corrections at each order. The first-order equation is one equation with two unknowns. Since the solutions to the unperturbed Schrddinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrddinger perturbation theory, and the equation in (4.32) becomes... 

In other words, the diagonal elements of the perturbing Hamiltonian provide the first-order correction to the energies of the spin manifold, and the nondiagonal elements give the second-order corrections. Perturbation theory also provides expressions for the calculation of the coefficients of the second-order corrected wavefunctions l / in terms of the original wavefunctions (p)... 

Perturbation theory also provides the natural mathematical framework for developing chemical concepts and explanations. Because the model H(0) corresponds to a simpler physical system that is presumably well understood, we can determine how the properties of the more complex system H evolve term by term from the perturbative corrections in Eq. (1.5a), and thereby elucidate how these properties originate from the terms contained in //(pertJ. For example, Eq. (1.5c) shows that the first-order correction E11 is merely the average (quantum-mechanical expectation value) of the perturbation H(pert) in the unperturbed eigenstate 0), a highly intuitive result. Most physical explanations in quantum mechanics can be traced back to this kind of perturbative reasoning, wherein the connection is drawn from what is well understood to the specific phenomenon of interest. 

It should be apparent that the expressions for the wave functions after interaction [equations (3.38) and (3.39)] are equivalent to the Rayleigh-Schrodinger perturbation theory (RSPT) result for the perturbed wave function correct to first order [equation (A.109)]. Similarly, the parallel between the MO energies [equations (3.33) and (3.34)] and the RSPT energy correct to second order [equation (A. 110)] is obvious. The missing first-order correction emphasizes the correspondence of the first-order corrected wave function and the second-order corrected energy. Note that equations (3.33), (3.34), (3.38), and (3.39) are valid under the same conditions required for the application of perturbation theory, namely that the perturbation be weak compared to energy differences. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

As an example of the connection between perturbation theory wave function corrections and polarizability, we now calculate the linear polarizability, ax. The states are corrected to first order in H. Since the polarization operator (Zx) is field independent, polarization terms linear in the electric field arise from products of the unperturbed states and their first-order corrections from the dipole operator. The corrected states are [12]... 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

We calculate the effects of the Hamiltonian (8.105) on these zeroth-order states using perturbation theory. This is exactly the same procedure as that which we used to construct the effective Hamiltonian in chapter 7. Our objective here is to formulate the terms in the effective Hamiltonian which describe the nuclear spin-rotation interaction and the susceptibility and chemical shift terms in the Zeeman Hamiltonian. We deal with them in much more detail at this point so that we can interpret the measurements on closed shell molecules by molecular beam magnetic resonance. The first-order corrections of the perturbation Hamiltonian are readily calculated to be... 

The SOC paths explicitly treated above have important consequences on the photophysical properties of the substates of a 3MLCT state. Perturbation theory can be used to illustrate the effects in the simple 4-orbital model as introduced above in Fig. 12. When taking into account significant SOC of the 3(dit )+1 substate with only the (d n ) state, as outlined above, the first-order corrected wave function 13(d7T ) j s°c can t>e written as (compare also [113])... 

Gp p, p r) at p = 0 have only simple poles and cuts as function of r- In perturbation theory this property holds true for the zero-order tcum, a simple pole being found for r = 0, The first order correction, however, at T = 0 has a pole of second order, due to the two factors of Gqj (p,t). This signals that the interaction wants to shift the pole according to the mechanism... 

Notice that this U(0) operator is not unitary Thus, in order to maintain the eigenvalues of the first-order corrected H. f / in Eq. 88 this operator has to be modified and made unitary before it can be inserted in Eq. 82a. This can be accomplished by orthonormalisation of the columns of the U(0) matrix, as is done in standard perturbation theory, or by maintaining all terms oi Dp and defining... 

Changa et al. [85] used the perturbation theory to estimate the energy eigenvalues of some low-lying electronic states. Their formula for the first order correction is a sum of terms of generalized hypergeometric functions. They also performed a numerical calculation based on the finite difference method. 

To calculate the corrections to the HLA we may take advantage of the smallness of V/E0 explicitly, using perturbation theory with respect to the terms that do not conserve the number of particles. All the first-order corrections to the energies vanish since the perturbation has no diagonal matrix elements in the HLA eigenstate basis. The correction to the energy Es of the state s) due to the perturbation V is given by... 

As noted the integration over the potential in the VSCF equations is A — 1 dimensional and so it is clear that if N is larger than 4 or so the integration becomes extremely computer intensive. Further it is usually necessary to go beyond the VSCF description to obtain accurate energies. Gerber and co-workers use second-order perturbation theory to correct VSCF energies [20,21]. This approach uses the virtual states defined above to correct the VSCF energies. The first-order correction vanishes since VSCF is correct to first order, and the second-order correction is... 

Both types of SCF perturbation theory generate similar-looking expressions for the first-order corrections to the density matrix ... 

To gain some experience in the evaluation of perturbation contributions to j and Ej and to motivate an analysis of a fundamental weakness of the Buillouin-Wigner perturbation theory (BWPT), let us now consider a few examples. First, we evaluate the first-order correction to the energy that arises from the n = 0 term in Eq. (3.11) ... 

---